Brownovo gibanje

V članku motiviramo idejo Brownovega gibanja in podamo njegovo matematično definicijo. Definiramo osnovne pojme, kot so slučajni proces, trajektorija, prirastek, filtracija in čas ustavljanja. Predstavimo in dokažemo nekaj osnovnih lastnosti ter rezultatov, kot je simetričnost Brownovega gibanja, šibka lastnost Markova, krepka lastnost Markova, princip zracljenja in presenetljiv rezultat, da je trajektorija Brownovega gibanja neodvedljiva funkcija.

Brownian motion

In this paper we motivate the idea of Brownian motion and give its mathematical definition. We define basic concepts such as a stochastic process, sample path, increment, filtration and stopping time. We present and prove some basic properties and results, such as the symmetry of Brownian motion, weak Markov property, strong Markov property, reflection principle and the surprising result that the sample path of Brownian motion is nowhere differentiable.